A210374 - OEIS

%I #11 Jul 16 2024 13:11:50

%S 0,4,19,40,68,104,149,204,270,348,439,544,664,800,953,1124,1314,1524,

%T 1755,2008,2284,2584,2909,3260,3638,4044,4479,4944,5440,5968,6529,

%U 7124,7754,8420,9123,9864,10644,11464,12325,13228,14174,15164

%N Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n+2.

%C A210374 is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3n-2.

%C See A210000 for a guide to related sequences.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).

%F Conjectures from _Colin Barker_, Dec 07 2017: (Start)

%F G.f.: x*(4 + 3*x - 12*x^2 + 6*x^3) / (1 - x )^4.

%F a(n) = (-36 + 47*n + 12*n^2 + n^3) / 6 for n>0.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.

%F (End)

%t a = 0; b = n; z1 = 45;

%t t[n_] := t[n] = Flatten[Table[w + x + y + z, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]

%t c[n_, k_] := c[n, k] = Count[t[n], k]

%t Table[c[n, n + 2], {n, 0, z1}]    (* A210374 *)

%t Table[c[n, 3 n - 2], {n, 0, z1}]  (* A210374 *)

%Y Cf. A210000.

%K nonn

%O 0,2

%A _Clark Kimberling_, Mar 20 2012